Kathrin Klamroth

Integrating Approximation and Interactive Decision Making in Multicriteria Optimization

We present a new interactive hybrid approach for solving multicriteria optimization problems where features of approximation methods and interactive approaches are incorporated. We produce rough approximations of the nondominated set and let the decision maker indicate with the help of reference points where to refine the approximation. In this way, (s)he iteratively directs the search toward the best nondominated solution. After the decision maker has identified the most interesting region of the nondominated set, the final solution can be fine-tuned with existing interactive methods. We suggest different ways of updating the reference point as well as discuss visualizations that can be used in comparing different nondominated solutions. The new method is computationally relatively inexpensive and easy to use for the decision maker.

Single-facility location problems with barriers

Measuring Distances.- Shortest Paths in the Presence of Barriers.- Location Problems with Barriers-Basic Concepts and Literature Review.- Bounds for Location Problems with Barriers.- Planar Location Problems with Polyhedral Barriers.- Location Problems with a Circular Barrier.- Weber Problems with a Line Barrier.- Weber Problems with Block Norms.- Center Problems with the Manhattan Metric.- Multicriteria Location Problems with Polyhedral Barriers.- Location with Barriers Put to Work in Practice.

Connectedness of efficient solutions in multiple criteria combinatorial optimization

Abstract In multiple criteria optimization an important research topic is the topological structure of the set X e of efficient solutions. Of major interest is the connectedness of X e , since it would allow the determination of Xe without considering non-efficient solutions in the process. We review general results on the subject, including the connectedness result for efficient solutions in multiple criteria linear programming. This result can be used to derive a definition of connectedness for discrete optimization problems. We present a counterexample to a previously stated result in this area, namely that the set of efficient solutions of the shortest path problem is connected. We will also show that connectedness does not hold for another important problem in discrete multiple criteria optimization: the spanning tree problem.

An efficient solution method for Weber problems with barriers based on genetic algorithms

In this paper we consider the problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers. It is assumed that a barrier is a region where neither facility location nor travelling are permitted. The resulting non-convex optimization problem can be reduced to a finite series of convex subproblems, which can be solved by the Weiszfeld algorithm in case of the Weber objective function and Euclidean distances. A solution method is presented that, by iteratively executing a genetic algorithm for the selection of subproblems, quickly finds a solution of the global problem. Visibility arguments are used to reduce the number of subproblems that need to be considered, and numerical examples are presented.

Constrained optimization using multiple objective programming

In practical applications of mathematical programming it is frequently observed that the decision maker prefers apparently suboptimal solutions. A natural explanation for this phenomenon is that the applied mathematical model was not sufficiently realistic and did not fully represent all the decision makers criteria and constraints. Since multicriteria optimization approaches are specifically designed to incorporate such complex preference structures, they gain more and more importance in application areas as, for example, engineering design and capital budgeting. The aim of this paper is to analyze optimization problems both from a constrained programming and a multicriteria programming perspective. It is shown that both formulations share important properties, and that many classical solution approaches have correspondences in the respective models. The analysis naturally leads to a discussion of the applicability of some recent approximation techniques for multicriteria programming problems for the approximation of optimal solutions and of Lagrange multipliers in convex constrained programming. Convergence results are proven for convex and nonconvex problems.

A reduction result for location problems with polyhedral barriers

Abstract In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facilities where a set of polyhedral barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the median and center objective functions). An exact algorithm and a heuristic solution procedure based on this reduction result are developed.

Equivalence of balance points and Pareto solutions in multiple-objective programming

It is shown that the concept of balance points introduced by Galperin (Ref. 1) is equivalent to the concept of Pareto optimality.

The multi-facility location-allocation problem with polyhedral barriers

In this paper we consider the problem of locating N new facilities with respect to M existing facilities in the plane and in the presence of polyhedral barriers. We assume that a barrier is a region where neither facility location nor traveling is permitted. For the resulting multi-dimensional mixed-integer optimization problem two different alternate location and allocation procedures are developed. Numerical examples show the superiority of a joint treatment of all assignment variables, including those specifying the routes taken around the barrier polyhedra, over a separate iterative solution of the assignment problem and the single-facility location problems in the presence of barriers.

Norm-Based Approximation in Bicriteria Programming

An algorithm to approximate the nondominated set of continuous and discrete bicriteria programs is proposed. The algorithm employs block norms to find an approximation and evaluate its quality. By automatically adapting to the problems structure and scaling, the approximation is constructed objectively without interaction with the decision maker. Mathematical and practical examples are included.

Planar Weber location problems with barriers and block norms

The Weber problem for a given finite set of existing facilities Ex={Ex1,Ex2,...,ExM}⊂∝2 with positive weights wm (m=1,...,M) is to find a new facility X*∈∝2 such that Σm=1Mwmd(X,Exm) is minimized for some distance function d. In this paper we consider distances defined by block norms.A variation of this problem is obtained if barriers are introduced which are convex polyhedral subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers, like lakes, military regions, national parks or mountains, are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the presence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a discretization result: one of the grid points in the grid defined by the existing facilities and the fundamental directions of the polyhedral distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.